Stability conjecture for packed words in the plactic monoid

A word $u$ has maximum letter $m$ and is $m$-packed if it contains at least one copy of every integer in $[m]=\{1,\ldots,m\}$. A word $u$ is $m$-stable when its centralizers satisfy $C(u^k)=C(u^{k+1})$ for the relevant stability range $m\le k$. Stability conjecture. If $u$ is $m$-packed, then $u$ is $m$-stable. This is supported by computations for $m=3,4$ and words of length at most $8$; the cases $m=1$ and $m=2$ were already established, but the general assertion remains open.